B. Grammaticos and A. Ramani Dedicated to Martin D. Kruskal on the occasion of his 70th birthday ...

16 Pages·2016·0.42 MB·English

Methods and Applications of Analysis © 1997 International Press
4 (2) 1997, pp. 196-211 ISSN 1073-2772
RETRACING THE PAINLEVE-GAMBIER CLASSIFICATION FOR
DISCRETE SYSTEMS
B. Grammaticos and A. Ramani
Dedicated to Martin D. Kruskal on the occasion of his 70th birthday.
ABSTRACT. We present the discrete equivalent of the Painleve list of second order
differential equations which possess the Painleve property. In the discrete case,
the equations we consider are three-point rational mappings. The investigation
of their integrable character is based on the singularity confinement criterion.
The mappings we obtain belong to three large classes in perfect analogy with
the continuous case. The first class contains mappings integrable in terms of
elliptic functions. The second class contains the mappings that can be reduced
to linear ones. Finally, the third class consists of mappings that define discrete
transcendents, i.e., the discrete analogs of the Painleve equations.
1. Introduction
The integrability of low-order differential equations attracted the attention of late
nineteenth and early twentieth century mathematicians. Thanks to Painleve and his
school, considerable progress has been realized in this domain but, somehow, the initial
impetus gradually disappeared, and the whole integrability business faded out. Then,
through one of those miracles that do occur in the par excellence rationalistic domain
of mathematics and physics, the interest in integrability was revived, more than half
a century later.
While investigating a model of a lattice with nonlinear interactions, Fermi, Pasta,
and Ulam (FPU) [10] made an astonishing discovery: instead of energy equipartition
between the various modes, the system exhibited a recurrent behavior. This indicated
that some stable structure persisted in the evolution of the system. Several years
later, Kruskal and Zabusky [43] investigated an equation that could be interpreted as
a continuous limit of the FPU model and was known to possess a stable solitary-wave
solution: the Korteweg-deVries (KdV) equation [34]. A new astonishing discovery
was made at this point: the solitary waves of KdV were stable even when interacting
among themselves. This led Kruskal and his collaborators to conjecture (and, shortly
afterwards, to prove) the integrability of the KdV equation [13]. This integrability
turned out to be of an unusual type. While the integrable partial differential equations
(PDEs) known at the time were linearizable through a Cole-Hopf transformation, the
KdV equation was fundamentally different. As discovered by Kruskal et al., it could be
written as the compatibility condition of an overdetermined linear system. The latter
Received October...13, 1995, revised August 8, 1996.
1991 Mathematics Subject Classification: integrability, Painleve equations, discrete symmetry.
Key words and phrases: 35Q53, 58F01, 35Q58, 39A10, 65Q05.
196
THE PAINLEVE-GAMBIER CLASSIFICATION FOR DISCRETE SYSTEMS 197
was subsequently called the "Lax pair" [27], a notion which became a cornerstone of
the modern approach to integrability. The crucial step came with the realization that,
given the Lax pair, the techniques of inverse scattering (for the reconstruction of the
potential from scattering data) could be used in order to solve the initial-value problem
for KdV. This discovery rekindled the interest in integrable systems, an interest that
has not faded away more than a quarter century later.
It was natural in this framework of integrable systems to come back to the question
of the Painleve equations. Painleve [32] had, in fact, studied second-order differential
equations of the form w" = f(wf'wz) where / is rational in w', algebraic in w, and
1 1
analytic in z, from the point of view of the singularity structure of their solutions. He
had, with the help of Gambier [12], identified fifty equations the solutions of which
did not contain movable critical singularities. They have shown that the majority of
these equations were integrable (or reducible to some other, simpler, equation among
the fifty). However, six of these equations could not be integrated with the techniques
available at that time. Painleve's intuition led him to decide that they were integrable
indeed, defining new transcendents. The derivation of "Lax pairs" for the Painleve
equations did not take too long. Garnier [14] derived the linear systems that led, as
compatibility conditions, to the Painleve equations, but did not use them in order to
integrate the latter. This was done for the first time by Ablowitz and Segur [4], using
inverse scattering techniques. They have shown in fact that the linearization of the
Painleve equations can be realized through integrodifferential equations.
Ablowitz and Segur (in close collaboration with Kruskal) [3], and also Hastings and
McLeod [22], made another observation. All the reductions of integrable PDEs that
could be written as second-order ordinary differential equations (ODEs) were members
of the Painleve-Gambier list. This led naturally to the formulation in collaboration
with one of us (Ramani) of the ARS conjecture [2]: "Every nonlinear ODE obtained
by an exact reduction of a nonlinear PDE solvable by inverse scattering techniques is
of P-type". This means that all its solutions have the Painleve property, i.e., they are
free of movable critical singularities. They also developed an efficient algorithm for
the investigation of the Painleve property. This led to a whole branch in the domain
of integrability [38] and, what is perhaps even more important, it forged a tool, both
efficient and easy to use, for the study of integrability.
While continuous systems were in full development, the situation was quite different
for discrete systems. One of the reasons (perhaps the main one) was that there was,
at the time, not much "physical" interest in discrete systems. The modeling of the
various physical phenomena proceeded usually through continuous systems, and on
those occasions where discrete systems were considered, this was done solely in order
to study their chaotic behavior. Another, non-negligible, point was the fact that dis-
crete systems are, a priori, much more difficult to study than continuous ones. (This
last statement is to be understood in the situation of the 70s and early 80s when the
tools for the study of discrete systems had not yet been developed). Isolated efforts,
of course, did lead to very interesting results in the domain of discrete integrability.
We can mention here the works of Ablowitz and Ladik [1], Hirota [23], and Capel
and collaborators [35], but, on the whole, those attempts did not manage to create
an impetus in the right direction. What also was lacking was the equivalent of the
ARS-Painleve approach: an integrability detector for discrete systems. The latter had
to wait till the 90s. At that time, by a happy coincidence that occurs frequently in
198 GRAMMATICOS AND RAMANI
scientific research, the conditions became right for the derivation of a discrete inte-
grability detector. (Note that "coincidence" is perhaps not the right word. What
really happens is a collective phenomenon whereby a critical mass is reached and the
proper impetus is maintained). First, recent work in field theory had turned the in-
terest towards discrete systems [6]: physicists started realizing that discrete systems
were more fundamental than continuous ones. Second, integrable discrete systems
had started appearing [31], replacing the previous scarcity by a body of examples that
would allow the formulation and testing of conjectures on integrability. It was in this
context that we proposed [20] the "singularity confinement" method, that has (four
years later) proven beyond doubt its efficiency as an integrability detector. The prin-
ciple is in fact simple. In a rational mapping, singularities may appear spontaneously
due to a particular choice of initial condition. In analogy to the continuous case, we
call these singularities "movable". Our conjecture states that, in integrable systems,
these singularities must disappear after a few iterations. This is what is meant by
"confinement". The implementation of this method has been reviewed in detail in
[41]. We shall limit ourselves here to probably the most important application of the
singularity confinement method: the derivation of the discrete Painleve equations.
As was mentioned above, Painleve and Gambier accomplished the difficult task
of classifying all the integrable second-order equations of the form w" = f(wf,w,z).
These equations turned out to be of the utmost usefulness in the study of physical
problems. It was thus natural to try to establish the analogous results in the case of
discrete equations. Discrete Painleve equations had already made their appearance in
the studies related to the Ising model [25, 28], and, in fact, the first reference remarked
on the relation of the discrete equation obtained and Painleve equations. Similarly,
in a work of Jimbo, Miwa, and Ueno [26] going back to 1981, a discrete Painleve
equation appears (it is the "alternate" d-Pi of [11]), but its analogy to the continuous
Painleve equation was not established. A more recent and unambiguous reference
to a discrete Painleve equation is made in the work of Brezin and Kazakov [6] who,
studying a two-dimensional model of quantum gravity, obtained the 'standard' form
of d-Pj. Shortly afterwards, d-Pn was also obtained [30, 33], and the introduction of
singularity confinement allowed us to derive [39] the remaining d-Ps up to d-Py. Since
this first work, many more results have been obtained, and today we can show that
the whole Painleve-Gambier classification can be retraced in a discrete setting. Before
proceeding further, we must point out that our approach is not the same as that of
Painleve and Gambier. Those authors investigated systematically (Painleve with his
"a-method" while Gambier used Kovalevskaya's approach) second-order ODEs and
identified those which satisfied the integrability criterion. In our approach, we do not
implement singularity confinement in the most general discrete system but, rather,
look for the discrete analogs of the equations of the Painleve-Gambier list. Thus our
search is guided by the results of the continuous case, and the singularity confinement
criterion is used in order to ensure integrability of the forms obtained.
The discrete equivalent to the second-order differential equations of the Painleve
family we are going to study here is the three-point explicit rational mapping. It
is clear that this is not the only possibility. However, it is a most natural one and
as we shall show, it is rich enough to yield the expected results. Before examining
the three-point case, we shall start with two-point mappings which are equivalent to
first-order differential equations.
THE PAINLEVE-GAMBIER CLASSIFICATION FOR DISCRETE SYSTEMS 199
2. First-order equations and two-point mappings
The study of integrable differential equations was initiated by Fuchs who studied first-
order equations, (algebraic in w w') the solutions of which have fixed critical points.
:
Poincare has shown that these equations either have all their solutions algebraic, or
are algebraically reducible to a quadrature, or are of Riccati form. If we focus on
equations explicit in the first derivative, w' = f{w,z), the only first-order equations
that have what later became known as the Painleve property are
w' = aw + 6, (2.1)
w' = -w2 + aw + 6. (2.2)
It is instructive to analyze the first-order equations in the light of modern singularity
analysis. Let us start with a first-order equation wf = f(w,z) where / is a polynomial
in w of degree k. Then the dominant singular behavior of the solutions of the equations
is w oc (z—zo)~1^k~'1\ Thus, except for the Riccati case k = 2, the singularity presents
a branching incompatible with the Painleve property.
What are the discrete equivalents of equations (2.1) and (2.2)? Clearly, the linear
map
Xn+l = UXn + b (2.3)
is the discrete analog of (2.1). In the case of the Riccati equations, the discretization
of the derivative does not cause any difficulty: a finite difference x +i — x suffices.
n n
The difficulty lies in the discretization of the nonlinear term. To make a long story
short [40], the only discretization that leads to an integrable mapping is to replace w2
by x +ix . Thus, the discrete Riccati equation is the homographic mapping
n n
ax + b
n
x"+1 = ^Trf- (2-4)
Using scaling, we can bring (2.4) to the form £ i— x = (—x^ + ax + 6)/(l — aj ),
n+ n n n
a form that clearly evokes (2.2). The parallel between the continuous and the discrete
Riccati equations is perfect. Both are linearizable through a Cole-Hopf transformation.
Putting w = P/Q, or x = P/Q, allows one to reduce them to a linear equation of
second order or to a linear three-point mapping, respectively.
What does singularity confinement tell us about two-point mappings? Let us con-
sider in general x +i = f(x n) where / is rational in x . As was shown in [40], the
n ni n
only form that satisfies the singularity requirement is
/(a! n)=o+ (25)
*' ?(*.-W* -
with positive integers z/fc, provided that for all fc, fik ^ &• Indeed, if x = Pk at some
n
step, then x i diverges, x 2 = a, and x s is finite. So the mapping propagates
n+ n+ n+
without any further difficulty. However, if we consider the "backward" evolution,
then (2.5) solved for x in terms of x +i leads to multideterminacy, and the number
n n
of preimages grows exponentially with the number of 'backward' iterations. Indeed,
the only mapping of this form with no growth is just the homographic (2.4).
The argument of non-proliferation of preimages [21] is essential for the integrability
of discrete systems. In fact, in every case that we consider, this no-growth criterion
200 GRAMMATICOS AND RAMANI
is implemented even before one starts examining the possible singularities and their
confinement.
3. The second-order (Painleve/Gambier) equations
and the corresponding three-point mappings
The aim of this paper is to establish the discrete analog of the Painleve/Gambier
classification of integrable second-order differential equations. Before embarking upon
examining the discrete forms, one must establish clearly which are the continuous
equations one wishes to discretize. The common lore is that there exist 50 canonical
equations of the form
wr = f(w\w,z) (3.1)
(with / rational in w', algebraic in w, and analytic in z) that possess the Painleve
property. They were first compiled by Gambier [12] and canonized through the book
of Ince [24]. Recently, Cosgrove [9] raised objections as to the canonical nature of this
list and, in a work that is (unfortunately) still part of the apocrypha of integrability,
talked about 74, 81, or even 120 canonical forms. (In fact, Gambier himself has made
clear that the number of possible forms of the equations of the form (3.1) ranges well
into the hundreds). So, had it not been for Gambier, we would be faced with a hard
task.
Fortunately, in his seminal work, Gambier presented a table of 24 equations that
he considered as fundamental. In fact, Gambier constructs "... un tableau {t), dont
il suffit d'indiquer la nature de chaque equation pour en deduire immediatement la
nature de toutes les equations T et en donner Vintegration effective, ... ". In other
words, determining the solutions of the 24 equations of his table {t) would allow one
to find the solutions of all the equations of the larger list (T), which are the 50 listed
by Ince, and through a homographic transformation of the dependent variable, plus a
transformation of the independent variable, the solutions of all the equations of form
(3.1) with the Painleve property.
Here is the Gambier (t) list, where a, 6, c, d, e are constant, q and r are free
functions of z, and / , ^ , and ^n are given functions of q and r:
n n
w" = 0, (Gl)
w" = 6™2, (G2)
" = 6w2 - 1/24, (G3)
w
w" = Gw2 + z, (G4)
w" = -3ww' -w3 + q{w' + w2), (G5)
w" = -2ww'. + qw' + q'w, (G6)
w" = 2w3, (G7)
w" = 2w3 + aw + 6, (G8)
w" = 2w3 + zw + a, (G9)
THE PAINLEVE-GAMBIBR CLASSIFICATION FOR DISCRETE SYSTEMS 201
w'2
w" = —,. (G10)
w
/2 d
w" = — + aw2" + bw2 + c + —, (Gil)
^ = + -(aw2 + 6) + cuf + -, (G12)
.tu z z it;
/2 /
w" = — + a— - ^ 4- rtt;^' + r'w2, (G13)
«," = f i _ I) «„'2L +,w _ ^^^22 ^ j^^^z
+ w2 (G14)
V n) w H (n + 2)2 n + 2 v ^
w" = (1 - - ) — + (Uw + 4> - —- J «/
\ n) w \ n nw )
^ = ^_ + ^L + 4aw2 + 26w - £-, (G17)
2if; 2 2K;
t/;77 = — + -IT- + 4W2 + 2(z2 - a)w - —-, (G18)
2w 2 v ; 2^;' v ;
." = 4^, (G19)
" = '2 (-L + -J—) + ( - 1)2 ( h.) J^, (G21)
w w W aw + +cw +
\2w w — lj \ w) w — 1
w = ^ ( T" + 7 - — + ^ 2 ( aw + - ) + c- + —*— -Z, (G22)
\2t(; w — lj z z* \ wj z w — r\
w" = w' [—+ +
w w — 1 w — a
c d e
+ w{w-l){w-a)(b+^+ _ "^3 + ^ " ), (G23)
w2 ((w — I)2 (w — a)2
\iy ty — 1 w — zj \z z — 1 w — zj
«^( _- ^ ti&zi.).
+ 0 +c + (G24)
202 GRAMMATICOS AND RAMANI
First, let us remark that two of these equations, namely (Gl) and (G6) are just the
derivatives of the equations we encountered at first-order. Equation (Gl) is obviously
the derivative of a linear equation, and (G6) is the derivative of a Riccati equation.
The remaining equations can be regrouped into three classes corresponding to the
method of integration. The first class contains equations (2, 3, 7, 8, 10, 11, 16, 17, 19,
20, 21, 23). They are integrable through quadratures in terms of elliptic functions,
and the generic equation of this class is equation (G23). The second class contains
equations (5, 13, 14, 15) that can be linearized. All of them are generalizations of the
Riccati equation in some sense, and equation (G15) which is due to Gambier is the
generic one of this class. Finally, the third class (4, 9, 12, 18, 22, 24) corresponds to the
transcendental equations of Painleve, Pi to Pyi. These equations are not integrable
through quadratures, nor are they reducible to a linear differential system. Rather,
their integration is obtained through 1ST techniques which reduce them to linear
integrodifferential equations. The generic equation is Pvi (G24): it can be reduced to
the other transcendents through the appropriate limits.
Let us now turn to the discrete forms of these equations. Concerning the derivatives
of the linear and Riccati equations, the discrete form is straightforward. We find, after
several simplifications in the case of the discrete Riccati equation
xn-\-l + ^n-l = a&n + &j (3-2)
. &n—l^n—1 / \
#n+l = #71-1 + a>n • {0o.d0)
xn
The remaining three classes have their discrete equivalent in perfect analogy to the
continuous result. Let us examine the class of equations solvable in terms of elliptic
functions first.
3.1. The elliptic-solutions mappings. The elliptic-solutions mappings have been
analyzed in [36] where the general form has been given. The QRT form we are going
to work with is
where the fas are quartic polynomials obtained in the following way. First, one defines
two symmetric matrices AQ and Ai of the form
(Oii Pi TA
ft e; Ci (3-5)
7< Ci ViJ
and the vector X = (x2 x l) . We obtain next V = AQX and W = AiX and
compute F = (/i /2 fo) through F = V x W.
Clearly, replacing AQ? AI by two independent linear combinations of AQ, AI will
leave the final result unchanged. It then always is possible to take ai = 0, and we,
generically, can put ao=l. (Non-generic cases, of course, may exist corresponding
to ao=0). Moreover, we always can take //o=0 and, generically, (JLI=1. Thus, from
the 12 parameters that were present in the QRT parametrization, only 8 survive at
this stage. Next, from the form of the mapping, it is clear that it is invariant under
homographic transformations of x. Since a homographic transformation has three
free parameters, one expects that their use will further help to simplify the mapping
THE PAINLEVE-GAMBIER CLASSIFICATION FOR DISCRETE SYSTEMS 203
by fixing three of the 8 remaining QRT parameters, bringing the result down to 5.
These transformations tend to be particularly intricate if performed directly. However,
singularity confinement can be of great help at this point. As we have explained in [40]
(see also [37]), the conditions for the appearance of a singularity of the QRT mapping
(3.4) is
/i(«(4))/3(*(i))-/22(^(0) = 0. (3.6)
This means that there exist 8 zeroes ajW's of (3.6) that lead to a singularity (loss of
a degree of freedom). If x = tfW, then x +i is independent of x -i and its value is
n n n
given by
M-hW-MW) 3
( 7)
The confinement of the singularity, with the assumption of a generic behavior, is
realized in a single step. This means (since the QRT mapping, being integrable, must
be confining) that each of the yW's calculated above is one of the roots x^ in order
to lead to an indeterminate form 0/0 and recovery of the lost degree of freedom.
Thus, a;M gives y^ through (3.7), which is equal to x^ for some j. It turns out
that the specific form of the QRT map ensures that y^ is precisely x^\ Thus, the
8 roots of (3.6) are grouped into four pairs. One enters the singularity through a?M
and exits through, say, x^+4:\ Once the roots are assigned in pairs, one can perform
the homographic transformation to bring three of the roots (say x^, x^2\ and x^)
to the values 0, 1, and oo respectively. Putting
x - x& a?(2) - xM
does just that. In fact, it is possible to go further and express all the parameters of
the QRT mapping in terms of the remaining sc^'s. As we have verified in [37], the
specific QRT mapping is the only one of the form (3.4) with the polynomials /^s of
degree less than 6 that satisfies the singularity confinement criterion. We are not going
to go through this analysis, nor give the cumbersome final expressions. It is perhaps
more interesting to display a case of the QRT mapping which goes over to (G23) in
the continuous limit {e —> 0). The corresponding matrices are:
A = 0 eo Co , Ai = 1 -2(a +1) a (3.8)
0
with eo = 2(e2(e + d + c + b(a2 + a + 1)) - 1) and Co = -e2{e + ad +
(a + l)(c + ab)). Although not singled out as a distinct equation in Gambler's list,
one case is particularly interesting since it leads to a very simple expression. Taking
6 = c=:d = e = 0in (G23) leads to the equation w" = wf2(^ + ^ + ^). We can
easily show that the discrete form of this equation is just
+ *,_! - 2* = (a;"+1 - *»)<*" - X^ (J- + -JL- + -J-) . (3.9)
gB+1 n
■" V •£«. Xn 1 x<n. a I
204 GRAMMATICOS AND RAMANI
As was shown in [36], the QRT mapping is integrable. Its first integral is given by the
correspondence
(ao + Kai)xl xl + (fo + Kp^Xn+^nixn+i + x )
+1 n
+ (70 + -KriXsn+i + xl) + (eo + Ke^Xn+^n
+ (Co + KCi)(x + a: ) + (/io + K/n) = 0 (3.10)
n+1 n
where K is the integration constant. Expression (3.10) further can be parametrized in
terms of elliptic functions [5]. This means that the solution of (3.10) is just a sampling
of an elliptic function over a discretized mesh, i.e., the solution is essentially the same
in the continuous and discrete cases. As we shall see in what follows, this will not be
the case for the remaining Painleve-Gambier mappings.
Once the generic mapping corresponding to (G23) is derived, it is easy to obtain
the nongeneric forms corresponding to the remaining equations of this class. We shall
not go into this elementary exercise, but rather proceed to the second class, that of
linearizable mappings.
3.2. The linearizable mappings. In [40], we have presented a generalization of
the QRT mapping that makes possible the description of linearizable mappings. This
generalization reads
MXn) - Xn-lf2(Xn) ,„ - - x
a:n+1 = Uf \(x«. n)\— - ^X -ijT37(XZ~)\ W'11'
n n
where the fas are polynomials in x and with coefficients that depend explicitly on n. In
fact, just as in the case of the Painleve-Gambier equations, the linearizable mappings
are the only ones for which the n-dependence is not strictly fixed but, rather, can be
written in terms of free functions.
Instead of analyzing (3.11) in its full generality, we shall limit ourselves to the forms
that are the discrete analogs of the linearizable equations of the Gambier list. The
generic mapping of this kind, corresponding to Gambier's equation (G15), is given by
the coupling of a mapping # +i homographic in x (discrete Riccati equation) with
n n
coefficients linear in y
n
ax x y + pXn+xXn + 7#n+i2/n + Sx y + ex + CVn + Vxn+1 + 0 = 0 (3.12)
n+1 n n n n n
with a discrete Riccati equation for y
n
Vn+i = ;—• (3.13)
Eliminating y between (3.12) and (3.13), we find for x a mapping of the form (3.11)
with fis quadratic in x . (However, this mapping is not the most general one of this
n
form.) Since the fas are quadratic the singularity condition
/i(s(i))/3(*(i)) " Mx^Mx®) = 0 (3.14)
has four roots a^1), x^2\ x^\ and x^ at each n. As explained in [17], due to the
cascade-Riccati structure of the mapping, the basic singularity pattern corresponding
to this equation is the following. When x becomes x^ at step n, the mapping loses
one degree of freedom and recovers it at the very next step through the value x^
at step n + 1. The same applies to the pair x^ and x^4K These are not the only
singularities of the mapping. But they will suffice in order to give the general form.
THE PAINLEVE-GAMBIER CLASSIFICATION FOR DISCRETE SYSTEMS 205
Next, we use the invariance of (3.12) under a homographic transformation at each n
and assign to three of the Xi specific values, say, a^1) = 0, x^ = 1, and x^ = oo.
We find that the two singularity patterns {0,1} and {^(3),oo} are realized only if
the mapping has the form (taking generically 77 = 1)
f3x +ix + x - 1 + x 2—- 0. (3.15)
n n n+1 n =
72/n + l
Using the homographic freedom for y, we can bring (3.15) to the form
x„ = i^. (3.16)
+1
This is precisely the form studied in [17] where we analyzed the Gambier mapping
2/n+1 = „ J-I ' (3.17a)
dx y + cr
x +i = — n n (3.17b)
n 1 — axn
where a = 0 or 1. The coefficients, a(n), 6(n), c(n), and (i(n) are not all free. They are
constrained by the singularity confinement requirement related to singularity patterns
where one first encounters x^ and, N steps later, x^ or one first encounters x^
and, N' steps later, x^\
We shall not go into these technical details here. It suffices to say that equation
(3.17) is indeed the discrete form of Gambler's equation. Like the latter, it is built
up from two Riccati equations in cascade, and indeed its continuous limit is equation
(G15). Just as (G5, G13, G14) can be obtained as special limits of the Gambier
equation (G15), in the same way, their discrete analogs are obtained from (3.17). Let
us start with (G5). Taking a = b = 0 (together with cr = 1), we find
Xn-id(n - l)(c(n - l)d(n)a; + 1) + x - 1
= n n
n+1 x -id(n - 1) + x - 1
n n
Putting u = — 1 + 1/x we obtain
_ u (pu + ) r
Un i = n+1 n q +
where r = c(n — l)d(n)d(n — 1), p = d(n — 1) — 1, and q = d(n — 1)(1 + c(n)d(n — 1)).
The mapping (3.19) is precisely the one obtained in [40] where we investigated the 3-
point mappings linearizable through a Cole-Hopf-like transformation. The remaining
two discrete equations are obtained in a straightforward way. In the continuous case,
they correspond to one of the two Riccati equations becoming linear. Equation (G13)
is in fact a derivative of the Riccati equation (but different from (G6)). In the discrete
case, we find the system (where, because we now have an extra freedom, b can be
scaled to 1)
S/n+i = Vn + c, (3.20a)
dx y + G ,r>r>™\
Xn+i = -.—n —n —, (3.20b)
J. CLXfi
or an equivalent form in terms of x alone.